Find all functions $f \colon \Bbb R \rightarrow \Bbb R $ that satisfy.....

Question

I find the following problem interesting :

Find all functions $f \colon \Bbb R \rightarrow \Bbb R $ that satisfy the inequality
$f(x+y)+f(y+z)+f(z+x) \geq 3f(x+2y+3z)?$

How Can I tackle the problem? Any hints will be appreciated.

Answer 1

Plugging in $x=t,y=0,z=0$ yields $$f(0) \geq f(t), \tag{1}$$ and plugging in $x=t/2,y=t/2,z=-t/2$ yields $$f(t) \geq f(0). \tag{2}$$

Hence, from $(1)$ and $(2),$ we get $f(t)=f(0) \quad \forall t$,so $f$ must be constant.
Conversely, any constant function $f$ clearly satisfies the given condition.